Summing Boolean algebras

In this paper we will study some families and subalgebras F of P(N) that let us characterize the unconditional convergence of series through the weak convergence of subseries Sigma(iis an element ofA) x(i), A is an element of F. As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.